Wireless communications involves several forms of signal modulation prior to transmission through a mobile channel. Some examples of the type of processing involved in wireless modulation include temporal processing (e.g. spread spectrum), spectral processing (e.g. orthogonal frequency division multiplexing, i.e. OFDM) and spatial processing (e.g. space-time coding). This kind of processing occurs over a single dimension (e.g. time, frequency, or spatial), and is sometimes referred to as single-scale modulation. Usually the type of processing involved is oftentimes selected based on the type of mobile channel conditions experienced. For instance, a mobile channel that provides no diversity (e.g. single-path fading) is sometimes addressed using spatial processing. On the other hand, a mobile channel where multipath is experienced (e.g. frequency-selective fading) may be better handled using spectral modulation such as OFDM.
Multiscale modulation involves processing the signal over two dimensions, namely time and frequency. Therefore, the output of a multiscale modulator is indexed by both a temporal range and frequency bin. This type of signal conditioning has the potential to match to two dimensions of the wireless channel rather than one.
Multiscale modulation can be visualized using a time-frequency tiling diagram. A sample tiling, derived from (1), is depicted in FIG. 1. The time-frequency tiling of a waveform composed of a sinusoid at frequency f0 and impulse t0 in this example results in energy in all subbands in the time-frequency domain intersecting both f0 and t0. A time-frequency representation of a signal can be obtained using a wavelet decomposition.
Previous work has addressed the use of the wavelet decomposition in digital communications. For instance, Wornell (8) has developed the concept of fractal modulation for multiscale communication. Moreover, in works such as (9) and (10), an optimal wavelet decomposition is chosen to account for specific types of channel conditions or transmitter imperfections. A particular problem with much previous work in the area of multiscale communications is that the issue of equalization of multipath channels at the receiver is oftentimes not specifically addressed. This is most likely due to the difficulty in trying to adaptively equalize the channel in two dimensions. Thus it would be desirable to be able to match a particular wavelet to instantaneous channel conditions with a minimal amount of interaction (i.e. feedback) between the receiver and transmitter. However, previous work has not addressed taking advantage of compact realizations of large wavelet families so as to match a wavelet with wireless channel conditions based on the selection of one or more scalar values. In this work, based on well-known compact wavelet decompositions, a parameterized wavelet modulation method is developed in which parameters are chosen to best match the wireless channel conditions.
Wavelet decompositions are normally defined in the continuous domain, where the so-called scaling function φ(x) is first derived (2), (3),φ(x)=Σckφ(2x−k)   (1)where {ck} is a real sequence. The sequence {ck} is of even length and must satisfy the following:Σck=2Σckck+2m=2δ(m)   (2)
Another important characteristic of wavelets which determines the “smoothness”, or continuity of the sequence defined by {ck} is the number of vanishing moments. If the wavelet has M (M≧1) vanishing moments, then the following holds:Σ(−1)kkmck=0, m=0,1(M−1)   (3)
A corresponding wavelet can now be defined asψ(x)=Σdkψ(2x−k)   (4)wheredk=(−1)kc1-k   (5)
Thus the dilates and translations of the wavelet function form an orthonormal basis:{√{square root over (2j)}ψ(2jx−k)}  (6)
Since the wavelet has compact support, the sequence {ck} is of finite length, assume that the sequence length is 2 N. Then the discrete wavelet transform may be defined starting with the two equal length sequences {ck} and {dk}. These two sequences can also be thought of as filters; they collectively form a perfect reconstruction filter bank.
A parameterized construction for wavelet and scaling filters for arbitrary values of N with M≦N vanishing moments was proposed in references (3) and (4). Let us assume that for a value of N, the filter coefficients are now denoted as {ckN}. Given an N-length wavelet parameter set {αi}(−π≦αi<π,0≦i<N), the coefficients {ckN}are derived by the recursion
                                                                        c                0                0                            =                              1                                  2                                                                                                                        c                1                0                            =                              1                                  2                                                                                                                        c                k                n                            =                                                1                  2                                [                                                                            (                                                                        c                                                      k                            -                            2                                                                                n                            -                            1                                                                          +                                                  c                          k                                                      n                            -                            1                                                                                              )                                        ⁢                                          (                                              1                        +                                                  cos                          ⁢                                                                                                          ⁢                                                      α                                                          n                              -                              1                                                                                                                          )                                                        +                                                                                                                                          (                                                            c                                                                        2                          ⁢                                                      (                                                          n                              +                              1                                                        )                                                                          -                        k                        -                        1                                                                    n                        -                        1                                                              +                                          c                                                                        2                          ⁢                                                      (                                                          n                              +                              1                                                        )                                                                          -                        k                        -                        3                                                                    n                        -                        1                                                                              )                                ⁢                                                      (                                          -                      1                                        )                                    k                                ⁢                sin                ⁢                                                                  ⁢                                  α                                      n                    -                    1                                                              ]                                                          (        7        )            
The wavelet construct in (7) is restrictive in the sense that the parameter set cannot in general be defined on the [−π,π]N continuum and still yield a wavelet with at least one vanishing moment. Pollen proved however that wavelets can be defined on the continuum [−π,π]N for arbitrary N (at the cost of smoothness). To examine these types of constructs for a given N, let us define the filter bank matrix FN as
                              F          N                =                  [                                                                      c                  0                                                            …                                                                                                                      ⁢                                      c                                                                  2                        ⁢                        N                                            -                      1                                                                                                                                            d                  0                                                            …                                                              d                                                            2                      ⁢                      N                                        -                    1                                                                                ]                                    (        8        )            
Then the filter bank matrix for N=1 is
                                          F            1                    =                                    1                              2                                      ⁡                          [                                                                                          cos                      ⁢                                                                                          ⁢                                              α                        0                                                                                                                                                -                        sin                                            ⁢                                                                                          ⁢                                              α                        0                                                                                                                                                        sin                      ⁢                                                                                          ⁢                                              α                        0                                                                                                                        cos                      ⁢                                                                                          ⁢                                              α                        0                                                                                                        ]                                      ⁢                                                      (        9        )            
The matrix in (9) is also sometimes known as the Givens rotation matrix. Similarly, the filter bank matrix for N=2 is (6)
                              F          2                =                              1                          8                                ⁢                                                 [                                                                                          1                      -                                              cos                        ⁢                                                                                                  ⁢                                                  α                          0                                                                    +                                              sin                        ⁢                                                                                                  ⁢                                                  α                          0                                                                                                                                                1                      +                                              cos                        ⁢                                                                                                  ⁢                                                  α                          0                                                                    +                                              sin                        ⁢                                                                                                  ⁢                                                  α                          0                                                                                                                                                1                      +                                              cos                        ⁢                                                                                                  ⁢                                                  α                          0                                                                    -                                              sin                        ⁢                                                                                                  ⁢                                                  α                          0                                                                                                                                                1                      -                                              cos                        ⁢                                                                                                  ⁢                                                  α                          0                                                                    -                                              sin                        ⁢                                                                                                  ⁢                                                  α                          0                                                                                                                                                                                                        1                        -                                                  cos                          ⁢                                                                                                          ⁢                                                      α                            0                                                                          -                                                  sin                          ⁢                                                                                                          ⁢                                                      α                            0                                                                                              ⁢                                                                                                                                                                      -                                              (                                                  1                          +                                                      cos                            ⁢                                                                                                                  ⁢                                                          α                              0                                                                                -                                                      sin                            ⁢                                                                                                                  ⁢                                                          α                              0                                                                                                      )                                                                                                                        1                      +                                              cos                        ⁢                                                                                                  ⁢                                                  α                          0                                                                    +                                              sin                        ⁢                                                                                                  ⁢                                                  α                          0                                                                                                                                                -                                              (                                                  1                          -                                                      cos                            ⁢                                                                                                                  ⁢                                                          α                              0                                                                                +                                                      sin                            ⁢                                                                                                                  ⁢                                                          α                              0                                                                                                      )                                                                                                        ]                                                          (        10        )            
The filter bank expressions for N=2 and N=3 are also sometimes known as Pollen filters, due to the fact that Pollen first proposed these two representations 7). Similarly, the filter bank matrix F1 becomes the Haar matrix when α0=π/4 and F2 is the Daubechies 4-tap filter bank when α0=π/6. Although the filter selectivity improves with increasing N, this comes at the cost of having to determine a larger set of parameters to define the wavelet. This can be seen in the parameterized expression for {ck3}:
                                          c            0            3                    =                                    1              4                        ⁡                          [                                                                    (                                          1                      +                                              cos                        ⁢                                                                                                  ⁢                                                  α                          0                                                                    +                                              sin                        ⁢                                                                                                  ⁢                                                  α                          0                                                                                      )                                    ⁢                                      (                                          1                      -                                              cos                        ⁢                                                                                                  ⁢                                                  α                          1                                                                    -                                              sin                        ⁢                                                                                                  ⁢                                                  α                          1                                                                                      )                                                  +                                  2                  ⁢                  cos                  ⁢                                                                          ⁢                                      α                    0                                    ⁢                  sin                  ⁢                                                                          ⁢                                      α                    1                                                              ]                                      ⁢                                  ⁢                              c            1            3                    =                                    1              4                        ⁡                          [                                                                    (                                          1                      -                                              cos                        ⁢                                                                                                  ⁢                                                  α                          0                                                                    +                                              sin                        ⁢                                                                                                  ⁢                                                  α                          0                                                                                      )                                    ⁢                                      (                                          1                      +                                              cos                        ⁢                                                                                                  ⁢                                                  α                          1                                                                    -                                              sin                        ⁢                                                                                                  ⁢                                                  α                          1                                                                                      )                                                  -                                  2                  ⁢                  cos                  ⁢                                                                          ⁢                                      α                    0                                    ⁢                  sin                  ⁢                                                                          ⁢                                      α                    1                                                              ]                                      ⁢                                  ⁢                              c            2            3                    =                                    1              2                        ⁡                          [                              1                +                                  cos                  ⁡                                      (                                                                  α                        0                                            -                                              α                        1                                                              )                                                  +                                  sin                  ⁡                                      (                                                                  α                        0                                            -                                              α                        1                                                              )                                                              ]                                      ⁢                                  ⁢                              c            3            3                    =                                    1              2                        ⁡                          [                              1                +                                  cos                  ⁡                                      (                                                                  α                        0                                            -                                              α                        1                                                              )                                                  -                                  sin                  ⁡                                      (                                                                  α                        0                                            -                                              α                        1                                                              )                                                              ]                                      ⁢                                  ⁢                              c            4            3                    =                      1            -                          c              2              3                        -                          c              0              3                                      ⁢                                  ⁢                              c            5            3                    =                      1            -                          c              3              3                        -                          c              1              3                                                          (        11        )            
Now two parameters must be determined before specifying the filter bank. In fact, although filter selectivity improves with an increasing number of coefficients, the complexity involved in setting the necessary parameters to form the filter bank also increases.
The wavelet decomposition can now be specified in terms of series of filter banks and resampling stages. Given an input sequence ai(n), then the output sequence may be derived as per the processing depicted in FIG. 2.
In modern technology, the filter process is performed digitally with a computational system such as a general purpose computer or an integrated circuit adapted for digital signal processing.
The number of resampling stages in a wavelet decomposition is sometimes referred to as the number of dilations. This processing can also be represented as a transformation of the input sequence by a unitary matrix. Assume that the input sequence at time index i, ai(n) is of (even) length K (i.e. 0≦n<K) and we want to define a discrete wavelet transformation (DWT) matrix TK of size K by K for a particular filter bank matrix FN. Moreover, assume that there are P dilations desired in the transformation. Then using the cited construction, an iterative method for deriving the transformation matrix may be found. Defining the time scale index as 1(0≦1≦P), a K-by-K filter bank matrix can be defined for each time scale:
                                          C            l                    =                      [                                                                                                                                                                                                                  F                              N                                                                                                                                                                                                                                                                                    0                                                              2                                ⁢                                                                  x                                  ⁡                                                                      (                                                                          K                                      -                                                                              2                                        ⁢                                        N                                                                            +                                      1                                                                        )                                                                                                                                                                                                                                                                                          0                                                              2                                ⁢                                x2                                                                                                                                                                                                        F                                .                                                            N                                                                                                                                          0                                                              2                                ⁢                                                                  x                                  ⁡                                                                      (                                                                          K                                      -                                                                              2                                        ⁢                                        N                                                                            -                                      1                                                                        )                                                                                                                                                                                                                                                                                          0                                                              2                                ⁢                                x4                                                                                                                                                                        F                              N                                                                                                                                          0                                                              2                                ⁢                                                                  x                                  ⁡                                                                      (                                                                          K                                      -                                                                              2                                        ⁢                                        N                                                                            -                                      3                                                                        )                                                                                                                                                                                                                                                            ⋮                                                                                ⋰                                                                                ⋮                                                                                              }                                        ⁢                                          (                                              K                        -                        R                                            )                                        ⁢                    rows                                                                                                                                                                                            0                          RxR                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              I                          R                                                                                                                                                                                                                                                                                                                                                                                      ]                          ⁢                                  ⁢                  R          =                      {                                                            0                                                                      l                    =                    0                                                                                                                                          ∑                                              i                        =                        1                                            P                                        ⁢                                          K                                              2                        i                                                                                                                                  l                    >                    0                                                                                                          (        12        )            
In (12), 0m×n is the zero matrix of m rows by n columns, and IR is the identity matrix of R rows by R columns. For each dilation, a permutation matrix Pυ(1≦υ≦P) can be defined as well:
                                          P            v                    =                      [                                                                                                                                                                                    1                                                                                0                                                                                …                                                                                0                                                                                0                                                                                0                                                                                0                                                                                0                                                                                                                                0                                                                                0                                                                                1                                                                                0                                                                                …                                                                                0                                                                                0                                                                                0                                                                                                                                0                                                                                0                                                                                0                                                                                0                                                                                1                                                                                0                                                                                …                                                                                0                                                                                                                                ⋮                                                                                                                                                                                                                          ⋮                                                                                                                                                                                                                                                                                                                                                                    ⋮                                                                                                                                                                                                                          ⋮                                                                                              }                                        ⁢                                          (                                                                        K                          -                          R                                                2                                            )                                        ⁢                    rows                                                                                                                                                                                                                        0                                                                                1                                                                                0                                                                                …                                                                                0                                                                                0                                                                                0                                                                                0                                                                                                                                0                                                                                0                                                                                0                                                                                1                                                                                0                                                                                …                                                                                0                                                                                0                                                                                                                                0                                                                                0                                                                                0                                                                                0                                                                                0                                                                                1                                                                                …                                                                                0                                                                                                                                ⋮                                                                                                                                                                                                                          ⋮                                                                                                                                                                                                                                                                                                                                                                    ⋮                                                                                                                                                                                                                          ⋮                                                                                              }                                        ⁢                                          (                                                                        K                          -                          R                                                2                                            )                                        ⁢                    rows                                                                                                                                                                                            0                          RxR                                                                                                                                                                                                                                            I                          R                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            ]                          ⁢                                  ⁢                  R          =                      {                                                            0                                                                      l                    =                    0                                                                                                                                          ∑                                              i                        =                        1                                            P                                        ⁢                                          K                                              2                        i                                                                                                                                  l                    >                    0                                                                                                          (        13        )            
Thus, for P dilations, the unitary transform matrix TK (P) may be determined asTK(P)=CPPP . . . C1C1C0   (14)
This matrix may now be used to modulate an input vector ai=[ai(0) . . . ai(K−1)]T (i being the symbol index). Illustratively, the elements of the input signal vector ai are the coefficients on a set of basis functions that represent the data, e.g. speech.
Thus the output sequence of such a modulation can be formed as the result of a matrix-vector multiplication xi=TKT(P)ai. As described above, the single net matrix TK(P) represents P inner products of the wavelet with the data. If we assume that FN results from a compact realization of a wavelet, then we can also assume that TK(P) is a function of the wavelet parameter set {αi}, i.e. xi=TK(P,{αi})ai.
Related Art may be found described in the references listed immediately below:                (1) Vetterli, Martin and Jelena Kovacevic. Wavelets and Subband Coding. Englewood Cliffs, N.J.: Prentice-Hall Inc., 1995.        (2) Strang, Gilbert and Truong Nguyen. Wavelets and Filter Banks. Wellesley, Mass.: Wellesley-Cambridge Press, 1996.        (3) Zou, Hehong and Ahmed H. Tewfik. “Parameterization of Compactly Supported Orthonormal Wavelets.” IEEE Transactions on Signal Processing. Vol. 41. No. 3. March 1993. pp. 1428-1431.        (4) Schneid, J. and S. Pittner. “On the Parameterization of the Coefficients of Dilation Equations for Compactly Supported Wavelets.” Computing. Vol. 51. May 1993. pp. 165-173.        (5) Vidakovic, Brani. “Pollen Bases and Daubechies-Lagarias Algorithm in MATLAB.” Jacket's Wavelets website.        (6) Silva, Vitor and Luis de Sa. “Analytical Optimization of COF Filter Banks.” IEEE Transactions on Signal Processing. Vol. 44. No. 6. June 1996. pp. 1564-1568.        (7) Pollen, D. “Parameterization of Compactly Supported Wavelets.” Aware Inc. technical report AD890503. May 1989.        (8) Wornell, Gregory W. “Emerging Applications of Multirate Signal Processing and Wavelets in Digital Communications.” Proceedings of the IEEE. Vol. 84. No. 4. April 1996. pp. 586-603.        (9) Lindsey, Alan R. “Wavelet Packet Modulation for Orthogonally Multiplexed Communication.” IEEE Transactions on Signal Processing. Vol. 45. No. 5. May 1997. pp. 1336-1339.        (10) Wong, K. -M., Wu, J., Davidson, T. N. and Jin, Q “Wavelet Packet Division Multiplexing and Wavelet Packet Design under Timing Error Effects.” IEEE Transactions on Signal Processing. Vol. 45. No. 12. December 1997. pp. 2877-2890.        